Introduction to Homotopy Theory- PART 3: CELL COMPLEXES
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- čas přidán 6. 01. 2021
- Topics covered include cell complexes, generating cofibrations, and several lemmas regarding these things. Based on nLab's section of the same name (see below).
PLEASE leave any misconceptions I had or inaccuracies in my video in the comments, and I'll pin them! In particular, I thought I found several typos in the nLab page, and I corrected them in my video, but I might have made an error.
References:
**nLab
ncatlab.org/nlab/show/Introdu...
**animation library
github.com/3b1b/manim
Music:
► Artist Attribution
• Music By: "KaizanBlu"
• Track Name: "Remember (Extended Mix)"
• CZcams Track Link: bit.ly/31Ma5s0
• Spotify Track Link: spoti.fi/2NUH3xZ
Feels like I'm watching an NPR ASMR relax video xD
If I recall correctly, you mentioned in a previous comment that you're still a university student. How to you pick up the advanced topics you present in your videos? Do you just pick up a textbook (or other introductory material) and start reading?
I attempted to read through nLab several times before it (hopefully!) stuck. With regards to something like stable homotopy theory, nLab represents, perhaps, the highest level of abstraction. But I think that underscores the non-linear manner in which I learn math. I took an algebraic topology course, but it hardly touched homotopy theory. Nonetheless, I was exposed to the open question regarding the stable homotopy groups of spheres. And so I had a goal. I searched Google for a "roadmap to stable homotopy theory" or something of the sort. Various forums were extremely helpful in outlining the various areas I needed to learn, and in providing references. I also began to watch lectures on CZcams, and read a few advanced articles and books. The purpose again being to better understand what I needed to learn (i.e. what I didn't know) and better motivate my study. For example, not just learning about derived functors and Ext and Tor, but developing an intuition that explained the reason they showed up in the Adam's spectral sequence, a central tool in stable homotopy theory.
I found nLab early on, but couldn't appreciate it then. After better understanding introductory algebraic topology, and developing some categorical vocabulary, I am finally able to appreciate the significance of the abstraction presented in model categories. So build up to abstraction, and if I had to just say one more thing, it would be to not force a linear trajectory of learning onto yourself- the end goal is what keeps us going. Look ahead, and keep returning to fill in the gaps in your understanding.
I've always considered myself a mediocre mathematics student- but I truly believe that love for the subject is all one needs to keep going.
@@rooney5395 Personally I still think stuff in nLab is just too advanced for me (well, I just started learning algebraic topology in less than a year). Do you have any recommendation for things to consider before going to nLab?
@@valerianmp I still think Hatcher's AT is a great place to start, pretty much all of it is useful (particularly Chapter 4). nLab is sort of reconstructing homotopy theory with the language of category theory and model categories, and so for many the largest barrier is just the category theory. I personally just picked this up as I went. It's really just a matter of developing intuition, I believe math3ma has a good series of posts on it. A really popular next step would be Peter May's Concise and More Concise Algebraic Topology, which develops the theory with more formalism while still being accessible to those with little background, though I personally haven't gone through these. Hope this helps!
@@rooney5395 I'm currently working out on Hatcher and chapter 4 does indeed feels like introducing a lot of interesting stuff!
Please if you can help it, do not keep it at such a sophisticated pace. I tried to follow up but your voice and volume is too nice and sophisticated. Try and pump up the volume on your next one bro. I fell flat asleep trying to follow, it is not your fault, but your voice is way too nice for a lecture. Lol